Problem 48 - Chemistry The Science in Context

Question

At an elevation where the boiling point of water is \(93^{\circ} \mathrm{C}, 100.0 \mathrm{g}\) of water at \(30^{\circ} \mathrm{C}\) absorbs \(290.0 \mathrm{kJ}\) of energy from a mountain climber's stove. Is this amount of energy sufficient to heat the water to its boiling point?

Step-by-Step Solution

Answer

Answer: Yes, the absorbed energy is sufficient to heat the water to its boiling point.

Step 1: Identify the given values

We are given: - The initial temperature of water, \(T_1 = 30^{\circ} \mathrm{C}\) - The elevation boiling point of water, \(T_2 = 93^{\circ} \mathrm{C}\) - The mass of water, \(m = 100.0\,\mathrm{g}\) - The energy absorbed by the water, \(Q_{absorbed} = 290.0\,\mathrm{kJ} = 290,000\,\mathrm{J}\) - The specific heat capacity of water, \(c = 4.18\,\mathrm{J/g^{\circ}C}\)

Step 2: Calculate the change in temperature

To find the change in temperature, we'll subtract the initial temperature from the boiling point: \(\Delta T = T_2 - T_1 = 93^{\circ}\mathrm{C} - 30^{\circ}\mathrm{C} = 63^{\circ}\mathrm{C}\)

Step 3: Determine the energy required to heat the water to its boiling point

Now we'll use the specific heat capacity formula to find the required energy: \(Q_{required} = mc\Delta T\) \(Q_{required} = (100.0\,\mathrm{g})(4.18\,\mathrm{J/g^{\circ}C})(63^{\circ}\mathrm{C})\) \(Q_{required} = 26,334\,\mathrm{J}\)

Step 4: Compare the required and absorbed energy

Now we'll compare the required energy with the absorbed energy to determine if the absorbed energy is sufficient for heating the water to its boiling point: \(Q_{required} = 26,334\,\mathrm{J}\) \(Q_{absorbed} = 290,000\,\mathrm{J}\) Since \(Q_{absorbed} > Q_{required}\), the absorbed energy is sufficient to heat the water to its boiling point.

Key Concepts

Specific Heat Capacity

Every substance has a unique capacity for storing thermal energy known as specific heat capacity. This is the amount of heat needed to raise the temperature of one gram of the substance by one degree Celsius (or one kelvin). Represented by the symbol 'c', it's measured in joules per gram per degree Celsius ( \( \text{J/g}^{\circ}\text{C} \) ).

For water, the specific heat capacity is relatively high at 4.18 \(\text{J/g}^{\circ}\text{C}\), which means it takes a considerable amount of energy to raise its temperature. This high 'c' makes water an excellent medium for regulating temperature, hence its use in heating systems and in our bodies. In the textbook exercise, we use the specific heat capacity along with the mass of the water and the temperature change to calculate the energy required to heat the water.

Energy Absorption in Heating

When a substance absorbs energy, it generally heats up, and the temperature rises until the energy is transferred elsewhere or the substance changes state, such as from liquid to gas. This absorbed energy is not always apparent as a temperature increase, especially if a phase change is occurring, because energy can be used to break intermolecular bonds in a process rather than increasing kinetic energy.

In the case of heating water on a stove, the energy from the stove is transferred to the water molecules, increasing their kinetic energy, which we observe as an increase in water temperature. The exercise demonstrates that the water absorbs 290 kJ of energy from the stove, which is a substantial quantity for 100 grams of water, and helps us understand whether this energy is sufficient for the water to reach its boiling point at the given elevation.

Temperature Change Calculations

To calculate the change in temperature, we first need to know the initial and final temperatures. The equation is straightforward: \(\Delta T = T_2 - T_1\), where \(\Delta T\) represents the change in temperature, \(T_1\) is the initial temperature, and \(T_2\) is the final temperature. It's important in calculations to maintain consistent units, typically degrees Celsius for these kinds of problems.

In our exercise, the water's initial temperature is 30 degrees Celsius, and the final temperature, or boiling point at the given elevation, is 93 degrees Celsius. By substituting these values into the temperature change equation, we are able to calculate that the water must undergo a 63-degree Celsius temperature increase to reach its boiling point.